# Mathematical Foundations for Balancing the Payment System in the Trade Credit Market

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## Abstract

**:**

## 1. Introduction

#### 1.1. Interbank vs. Trade Credit Clearing

- interbank network: 2770
- interbank networks: 1960
- trade credit network: 64
- trade credit networks: 51

#### 1.2. Historical Context and Brief Literature Review

#### 1.3. Basic Concepts

- Circular —is a situation where individual payments can only be settled in a specific order. This situation is resolvable by reordering the payment queue.
- Gridlock—is a situation in which several payments cannot be settled individually but can be settled simultaneously. This situation is resolvable with multilateral off-set.
- Deadlock—is a situation where the individual payments can be made only by adding liquidity to at least one of the system participants.

#### 1.4. Overview of Paper

## 2. Materials and Methods

#### 2.1. Notation and Definitions

- An
**obligation network**is a directed graph where the nodes6 represent firms and the edges represent the obligations. Parallel edges are allowed to represent multiple obligations between two firms: - A
**nominal liability matrix**is a matrix representing total obligations or liabilities between firms. We will define special vectors to describe properties of the nominal liability matrix. - A
**payment system**is constructed by adding special-function nodes to the obligation network. These special nodes represent sources of funds and a store of value. They can have connections to all nodes in the obligation network, and the set of all connections for each special node is expressed as a vector.

#### 2.1.1. Obligation Network

#### 2.1.2. Nominal Liability Matrix

#### 2.1.3. Balanced Net Position Vector

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Obligation-Clearing with a Liquidity Source

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 2.3. Simple Examples

#### 2.3.1. Obligation Chain

#### 2.3.2. Obligation Cycle

#### 2.3.3. Small Obligation Network with a Chain and a Cycle

## 3. Results

#### 3.1. General Formulation

#### 3.1.1. A Cycle as a Balanced Payment Subsystem

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.1.2. Finding the Maximum-Weight Set of Cycles

**Definition**

**8.**

**Problem**

**1.**

**Definition**

**9.**

#### 3.1.3. The Minimum-Cost Flow Problem

- Define the object function as a Grandsum function $\mu :{\mathbb{R}}^{{n}^{2}}\to \mathbb{R}$, which is the sum of all the elements of a given square $n\times n$ matrix. Looking for the minimum of the function $\mu \left(M\right)$ is equivalent to looking for the minimum-weight sub-network ${\mathcal{G}}_{m}$.
- Make sure that the payment system $(M,\mathbf{f})$ is balanced. The constraint above ensures this since $(L,\mathbf{f})$ is balanced by construction. In fact, since $\mathbf{f}=-\mathbf{b}$ and M uses the same cashflow vector $\mathbf{f}$, ensuring that M has the same net position vector $\mathbf{b}$ is enough to guarantee that $(M,\mathbf{f})$ will be balanced too.
- Make sure we are not introducing edges between nodes in sub-network ${\mathcal{G}}_{m}$ that do not exist in the obligation network $\mathcal{G}$. Therefore, all matrix elements ${M}_{ij}$ must have a value between 0 and ${L}_{ij}$.

**Problem**

**2.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Proof.**

#### 3.2. Using Balanced Payment Subsystems in the Trade Credit Market

**Theorem**

**6.**

**Proof.**

- Collect obligations to form an obligation network $\mathcal{G}$.
- Form a nominal liability matrix L and a payment system $(L,\mathbf{0})$ without external financing.
- Find a maximum-weight balanced payment subsystem T.
- Discharge the obligations in the balanced payment subsystem $(T,\mathbf{0})$ by sending set-off notices to all pertinent firms.
- Subtract the balanced payment subsystem T, such that $L-T=M$.
- Leave the remaining obligations in the nominal obligation matrix M to be discharged using the normal bank payment system.

## 4. Discussion: Practical Trade Credit Formulation

#### 4.1. Formal Model for Single Liquidity Source

- ${\mathbf{f}}_{1}$: cashflows from firms’ bank accounts (only positive balances used)
- ${\mathbf{f}}_{2}$: overdraft cashflows up to firms’ credit lines
- ${\mathbf{f}}_{3}$: cashflows to repay firms’ overdrafts (different set of firms from ${\mathbf{f}}_{\mathbf{2}}$’s)
- ${\mathbf{f}}_{4}$: cashflows into firms’ bank accounts (different set of firms from ${\mathbf{f}}_{\mathbf{1}}$’s),

- After all the obligations have been uploaded into the repository (which could be a blockchain), an obligation network exists, but the net positions are not yet known. The network is the first output, and the input to Step 2.
- Based on the network, the net positions are calculated, i.e., the components of vector $\mathbf{b}$. This is the output of Step 2 and the input to Step 3.
- The cyclic structure is determined. This is the output of Step 3 and the input to Step 4.
- The cyclic structure is removed from the obligation network, i.e., multilateral set-off is performed. This results in a new, acyclic obligation network with obligation amounts that, usually, are the same as before for a subset of firms, smaller for another subset, and zero for a third subset, which is the complement of the first two.
- The TCT multilateral set-off process is completed when set-off notices are sent to all the firms instructing them about what is left to pay.

#### 4.2. Generalization to Multiple Sources

#### 4.3. Optimizing the Use of Available Liquidity

**Theorem**

**7.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Central Bank Digital Currency. |

2 | Efficiency could be loosely defined as the ratio of debt cleared to the total initial debt. Due to network effects (probably factorial) that are not yet well-understood and that will be explored in future work, the injection of liquidity increases this value even after the liquidity used is subtracted from the debt cleared. |

3 | https://ec.europa.eu/cefdigital/wiki/display/CEFDIGITAL/eInvoicing+in+Italy (accessed on 18 September 2021). |

4 | In Slovenia, it is an institution equivalent to the UK’s Companies House. |

5 | Such a path usually involves multiple, and different, firms or ‘hops’ in each direction. |

6 | Usually referred to as vertices in graph theory. |

7 | We note that this is opposite to the ‘balance’ function $\delta \left(v\right)$ in Simić and Milanović (1992), which is defined using the standard fluid mechanics convention of ‘net flow = outflow − inflow’. In the present analysis, it is more convenient to follow financial intuition to define the net position as ‘inflow − outflow’. The only difference between Simić and Milanović (1992), and our analysis as far as this function is concerned, therefore, is merely a sign. |

8 | We should clarify that we are using the term ‘cashflow’ in a more general sense than its normal application in business, i.e., as the revenue per unit time of a given company. In this paper, cashflow means literally the movement or “flow” of currency over one or more hops of the network, i.e., between two or more companies. Such flow can also be a closed loop or cycle, and one of the nodes can also be a bank or other account-holding institution. The units are still $\left[\frac{\mathrm{currency}}{\mathrm{unit}\phantom{\rule{4.pt}{0ex}}\mathrm{time}}\right]$, but the time period is not important and can be trivially assumed to be 1, |

9 | There are many possible maximum-weight sets of cycles but only one maximum weight. Interestingly, depending on which maximum-weight set of cycles one finds, q could be different. Empirical tests on larger datasets show that the number of cycles can change within a range of about 1%. |

10 | In the context of a multigraph, there can be multiple edges between any two nodes. One might therefore expect different cycles to be associated with different edges between the same two nodes. While this is possible, it is not necessarily the case. The same edge can be the intersection of multiple cycles such that, if it happens to be the smallest weight of multiple cycles, subtracting one cycle will break all the others. |

11 | The ‘object function’ in optimization theory. |

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**Figure 2.**Payment system: obligation network, liquidity source/sink ${v}_{0}$, and vectors representing cashflows.

**Figure 8.**Example of gridlock resolution scenario. Sequence of steps depicted with dashed arrows is marked with letters from “a” to “e”.

**Figure 10.**Visualization of payment system with three liquidity sources (EUBOF 2021).

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**MDPI and ACS Style**

Fleischman, T.; Dini, P. Mathematical Foundations for Balancing the Payment System in the Trade Credit Market. *J. Risk Financial Manag.* **2021**, *14*, 452.
https://doi.org/10.3390/jrfm14090452

**AMA Style**

Fleischman T, Dini P. Mathematical Foundations for Balancing the Payment System in the Trade Credit Market. *Journal of Risk and Financial Management*. 2021; 14(9):452.
https://doi.org/10.3390/jrfm14090452

**Chicago/Turabian Style**

Fleischman, Tomaž, and Paolo Dini. 2021. "Mathematical Foundations for Balancing the Payment System in the Trade Credit Market" *Journal of Risk and Financial Management* 14, no. 9: 452.
https://doi.org/10.3390/jrfm14090452